Subdiffusive Valence Growth under Bounded Relational Flux: Structural Derivation of the Cascade Exponent β

The cascade exponent β, governing the power-law growth p(n) ~ n^β of the effective relational valence along the Cosmochrony relaxation cascade, was identified in a companion paper as the sole remaining structural postulate of the admissibility sub-programme. We show that the bounded-flux constraint |∂ₜ χᵥ| ≤ c_χ, inherited from Born–Infeld saturation, renders super-quadratic valence growth structurally impossible. Under a natural closure hypothesis relating valence to the cumulative exploration of the relational configuration space Ω_χ, the Cheeger isoperimetric inequality for Ramanujan–LPS graphs implies equation* p(n) c_ p(n), equation* from which it follows that p(n) C n², i.e.\ any admissible power-law exponent must satisfy β ≤ 2. By convention, the saturated growth rate p(n) ~ n² is identified with the maximal admissible regime; sub-saturated growth laws satisfying p(n) = o(n²) correspond to β < 2, and the physically relevant regime is β ∈ (0,1] (see Remark rem:beta-convention). This structural upper bound is derived without invoking phenomenological mass data and is therefore independent of the lepton or quark sector. The sharper confinement to the phenomenologically identified window β^* ∈ (0.09, 0.13) is not established here; we identify the additional geometric input required and formulate the residual problem precisely. The present result removes the last unconstrained degree of freedom of the O-series from the class of super-quadratic growth laws, consolidating the structural foundation of the programme.